Sec.2.2 The Physical Layer:Channels and Modems 47 since the pulse shape can be viewed as arising from filtering sample impulses.If the combination of the pulse-shaping filter at the input with the channel filter and equalizer filter has the (sinz)/z shape of Eq.(2.11)(or,equivalently,a combined frequency response equal to the ideal low-pass filter response),the output samples will re-create the input samples. The ideal low-pass filter and the (sin )/a pulse shape suggested by the sampling theorem are nice theoretically but not very practical.What is required in practice is a more realistic filter for which the sample values at the output replicate those at the input (i.e.,no intersymbol interference)while the high-frequency noise is filtered out. An elegant solution to this problem was given in a classic paper by Nyquist [Nyq28].He showed that intersymbol interference is avoided by using a filter H'(f) with odd symmetry at the band edge;that is,H'(f+W)=1-H'(f-W)for If<W, and H'(f)=0 for f>2W (see Fig.2.7).The filter H'(f)here is the composite of the pulse-shaping filter at the transmitter,the channel filter,and the equalizer filter.In prac- tice,such a filter usually cuts off rather sharply around f=W to avoid high-frequency noise,but ideal filtering is not required,thereby allowing considerable design flexibility. It is important to recognize that the sampling theorem specifies the number of samples per second that can be utilized on a low-pass channel,but it does not specify how many bits can be mapped into one sample.For example,two bits per sample could be achieved by the mapping 11-3,10-1,00--1,and 01--3.As discussed later,it is the noise that limits the number of bits per sample. 2.2.4 Bandpass Channels So far,we have considered only low-pass channels for which H(f)is large only for a frequency band around f =0.Most physical channels do not fall into this category, and instead have the property that (H(f)is significantly nonzero only within some frequency band f<If<f2,where f>0.These channels are called bandpass channels and many of them have the property that H(0)=0.A channel or waveform with H(0)=0 is said to have no dc component.From Eq.(2.3),it can be seen that this implies thathdt.The impulse response for these channels fluctuates H'{f) 1-H'{f'+W) H'(f'-W) -W 0 W Figure 2.7 Frequency response H(f)that satisfies the Nyquist criterion for no inter- symbol interference.Note that the response has odd symmetry around the point f=W.Sec. 2.2 The Physical Layer: Channels and Modems 47 since the pulse shape can be viewed as arising from filtering sample impulses. If the combination of the pulse-shaping filter at the input with the channel filter and equalizer filter has the (sinx)/x shape of Eq. (2.11) (or, equivalently, a combined frequency response equal to the ideal low-pass filter response), the output samples will re-create the input samples. The ideal low-pass filter and the (sinx)/x pulse shape suggested by the sampling theorem are nice theoretically but not very practical. What is required in practice is a more realistic filter for which the sample values at the output replicate those at the input (i.e., no intersymbol interference) while the high-frequency noise is filtered out. An elegant solution to this problem was given in a classic paper by Nyquist [Nyq28]. He showed that intersymbol interference is avoided by using a filter H'(f) with odd symmetry at the band edge; that is, H'(j +W) = 1 - H'(j - W) for IfI W, and H'(j) = 0 for IfI > 2W (see Fig. 2.7). The filter H'(j) here is the composite of the pulse-shaping filter at the transmitter, the channel filter, and the equalizer filter. In prac￾tice, such a filter usually cuts off rather sharply around f = W to avoid high-frequency noise, but ideal filtering is not required, thereby allowing considerable design flexibility. It is important to recognize that the sampling theorem specifies the number of samples per second that can be utilized on a low-pass channel, but it does not specify how many bits can be mapped into one sample. For example, two bits per sample could be achieved by the mapping 11 -* 3, 10 -* 1,00 -* -1, and 01 -* -3. As discussed later, it is the noise that limits the number of bits per sample. 2.2.4 Bandpass Channels So far, we have considered only low-pass channels for which IH(j)1 is large only for a frequency band around f = O. Most physical channels do not fall into this category, and instead have the property that I(H(j)1 is significantly nonzero only within some frequency band f1 IfI 12, where h > O. These channels are called bandpass channels and many of them have the property that H(O) = O. A channel or waveform with H(O) = 0 is said to have no dc component. From Eq. (2.3), it can be seen that this implies that J~= h(t)dt = O. The impulse response for these channels fluctuates -W o f ---+- 1 - H'(f' + W) W Figure 2.7 Frequency response H'(j) that satisfies the Nyquist criterion for no inter￾symbol interference. Note that the response has odd symmetry around the point f = ~V